reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th29:
  for f1,f2 being complex-valued Function holds
  |.f1(#)f2.| = |.f1.|(#)|.f2.|
proof
  let f1,f2 be complex-valued Function;
  thus dom (|.f1 (#) f2.|) = dom (f1 (#) f2) by VALUED_1:def 11
    .= dom f1 /\ dom f2 by VALUED_1:def 4
    .= dom f1 /\ dom (|.f2.|) by VALUED_1:def 11
    .= dom (|.f1.|) /\ dom (|.f2.|) by VALUED_1:def 11
    .= dom (|.f1.|(#)|.f2.|) by VALUED_1:def 4;
    let c be object;
    assume c in dom (|.f1 (#) f2.|);
    then
A5: c in dom (f1 (#) f2) by VALUED_1:def 11;
    thus (|.(f1(#)f2).|).c = |.(f1(#)f2).c.| by VALUED_1:18
      .= |.((f1.c)) * ((f2.c)).| by A5,VALUED_1:def 4
      .= |.((f1.c)).| * |.((f2.c)).| by COMPLEX1:65
      .= ((|.f1.|).c) *( |.((f2.c)).|) by VALUED_1:18
      .= ((|.f1.|).c) * ((|.f2.|).c) by VALUED_1:18
      .= (|.f1.|(#)|.f2.|).c by VALUED_1:5;
end;
